Long one
Solve the given equations: ⎩ ⎨ ⎧ 3 1 x 2 y 2 − 7 y 4 − 1 1 2 x y + 6 4 = 0 x 2 − 7 x y + 4 y 2 + 8 = 0
The solutions of x and y are in the below given forms:
⎩ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎧ x = ± A x = ± i C B x = ± D x = ± i G F y = ± A y = ∓ i C B y = ± E y = ∓ i I H
Find A + B + C + D + E + F + G + H + I .
Notation: i = − 1 denotes the imaginary unit .
The answer is 85.
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We just need to homogenize the system of equation (meaning we will use the two equations to form a new equation which will have same algebraic degree in terms of x and y. For eg, the equation x+2y=0 is a homogeneous equation of degree 1 in terms of x and y, similarly x²+7xy=0 is homogeneous of degree 2. x²+xy-y=0 is not a homogeneous eqn. In gen for homogeneous eqn, replacing (,y) with (kx,ky) will not change the overall equation for all real k except zero.) We will use homogenization so that we can get ratio of x and y and use it solve our system of equations. x²-7xy+4y²+8=0 => 7xy-8=x²+4y²...(i) 31x²y²-7y⁴-112xy+64=0 =>-7y⁴-18x²y²+(49x²y²-112xy+64)=0 =>-7y⁴-18x²y²+(7xy-8)²=0 =>-7y⁴-18x²y²+(x²+4y²)²=0...using (i)
=> x⁴-10x²y²+9y⁴=0 => x/y = 1,-1,3,-3. Now if you just substitute x in terms of y in equation (i) you will get all solutions of our system of equations. After solving you will get the values as follows: 2,2,3,3,1,36,17,4,17.